A model of two-velocity particles transport in a porous medium

A one-dimensional flow of suspension with two types of solid particles moving with different velocities in a porous medium is considered. A mathematical model of deep bed filtration which generalizes the known equations of mass balance and particle capture kinetics for a flow of fluid with identical particles is developed. The exact solution is evaluated at the filter inlet and on the concentration front of fast suspended and retained particles, asymptotic solutions are provided in certain vicinities of these lines. A global asymptotic solution to the problem with a small limit deposit is constructed. The asymptotics rapidly converges to the numerical solution.     

Multi-fluid model of suspension filtration in a porous medium

In the framework of a three-fluid approach, a new model of suspension filtration in a porous medium is constructed with account for the formation of a dense packing of trapped particles with finite permeability and porosity. The following three continua are considered: the carrier fluid, the suspended particles, and the deposited particles. For a one-dimensional transient flow of suspension, a system of equations for the concentrations of the suspended and deposited particles, the suspension velocity, and the pressure is constructed. Two cases of the flow in a porous medium are considered: plane and radial. Numerical solution is found using a finite-difference method. Numerical calculations are shown to be in agreement with an analytical solution for the simplest case of filtration with a constant velocity and constant porosity and permeability. A comparison is performed with the classic filtration models for five sets of experimental data on the contamination of a porous sample. It is shown that near the inlet boundary, where an intense deposition of particles takes place, the new model describes the concentration profile of the deposited particles more accurately than the classical model.     

A stochastic model for filtration of particulate suspensions with incomplete pore plugging

A population balance model for particulate suspension transport with capture of particles by porous medium accounting for complete and incomplete plugging of pores by retained particles is derived. The model accounts for pore space accessibility, due to restriction on finite size particle movement through the overall pore space, and for particle flux reduction, due to transport of particles by the fraction of the overall flux. The novel feature of the model is the residual pore conductivity after the particle retention in the pore and the possibility of one pore to capture several particles. A closed system of governing stochastic equations determines the evolution of size distributions for suspended particles and pores. Its averaging results in the closed system of hydrodynamic equations accounting for permeability and porosity reduction due to plugging. The problem of deep bed filtration of a single particle size suspension through a single pore size medium where a pore can be completely plugged by two particles allows for an exact analytical solution. The phenomenological deep bed filtration model follows from the analytical solution.     

悬浮体力学——流体力学与胶体科学交叉的新兴学科

本文介绍了交叉新兴学科悬浮体力学的概貌.该学科发生发展在低Re数流体力学与胶体科学的交叉边缘处.它具有很广泛的应用价值,也有许多吸引人的重要理论课题.近年来在这领域中的研究工作规模十分巨大.对这学科有兴趣的人们而言,这是十分令人鼓舞的时期.在给出了本学科的基本特征和主要研究内容之后,对于相互作用着的低Re数、低Stk数、高Kn数悬浮粒子的运动学,对于在均匀悬浮体中的相对布朗扩散,对于非均匀单分散与多分散悬浮体中的绝对布朗扩散,对于稀释单分散和多分散悬浮体中的重力沉降,对于悬浮粒子的布朗碰并、重力碰并和纯变形场碰并,以及对于高Pcclet数与低Pcclet数下的悬浮体的有效粘性等问题,本文都作了简要的论述.     

Particle suspension in a simple shear flow

A hydrodynamic model describing the particle distribution over the cross-section of a finely dispersed flow is proposed. The model is constructed on the basis of notions concerning the diffusion of particles induced by their random displacements in the process of relative motion of neighboring layers at constant shear velocity. It is shown that the suspension capacity of the flow is large for small particles due to thermal fluctuations and for relatively large particles due to shear-induced particle pulsations. There are critical particle sizes for which the particles are suspended and transported by the flow less effectively than larger or smaller particles.Ekaterinburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 112–121, January–February, 1995.     

Fluidization of a bed of solid particles

As is known, fluidization of a bed of solid particles by liquid or gas filtration takes place for certain critical values of the parameters of the filtration regime. The determination of these critical values and the nature of the transition is of interest in connection with the development of fluidization technology in many branches of industry, and also in connection with certain other questions, among which we note the problem of the suspension of a sand plug in an oil well.The two-dimensional fluidization problem has been examined previously [1] as the problem of the limiting equilibrium of a medium which cannot withstand arbitrarily small tensile stresses. This model describes well the behavior of many types of bulk media encountered in practice. However, many cases lie beyond the limits of this model because of the presence of bonding forces between the particles. Bonding may be due to the adhesive forces which arise during the fluidization of fine powders [2, 3], and/or to magnetic and electrostatic forces [3, 4]. Another example is the agglomeration of particles during gas fluidization when small amounts of liquid are injected [5]; still another is the case in which sand particles are surrounded by thin films of oil when a sand plug is suspended in an oil well.In the present paper an extension of the results obtained in [1] is used to examine fluidization of a bed with account taken of the bonding forces between the particles. The two- and three-dimensional problems are studied.     

A Stochastic Model for Particulate Suspension Flow in Porous Media

A population balance model for a particulate suspension transport with size exclusion capture of particles by porous rock is derived. The model accounts for particle flux reduction and pore space accessibility due to restriction for large particles to move through smaller pores – a particle is captured by a smaller pore and passes through a larger pore. Analytical solutions are obtained for a uniform pore size medium, and also for a medium with small pore size variation. For both cases, the equations for averaged concentrations significantly differ from the classical deep bed filtration model.     

Exact solutions to the problem of deep-bed filtration with retardation of a jump in concentration within the framework of the nonlinear two-velocity model

Exact solutions with plane and cylindricalwaves are obtained for one-dimensional problems of injection of a suspension into a porous reservoir when lagging of the suspended particles behind the carrier fluid is taken into account in the case of large change in the porosity. It is shown that taking lagging of the particles behind the fluid into account can lead to slowing-down the motion of jump in concentration. This is in agreement with the results of a series of experiments. It is also noted that, in principle, models in which the particles pass in average ahead of the carrier fluid are possible in the problems of deep bed filtration.     

Modified Particle Detachment Model for Colloidal Transport in Porous Media

Particle detachment from the rock during suspension transport in porous media was widely observed in laboratory corefloods and for flows in natural reservoirs. A new mathematical model for detachment of particles is based on mechanical equilibrium of a particle positioned on the internal cake or matrix surface in the pore space. The torque balance of drag, electrostatic, lifting and gravity forces, acting on the particle from the matrix and the moving fluid, is considered. The torque balance determines maximum retention concentration during the particle capture. The particle torque equilibrium is determined by the dimensionless ratio between the drag and normal forces acting on the particle. The maximum retention function of the dimensionless ratio (dislodging number) closes system of governing equations for colloid transport with particle release. One-dimensional problem of coreflooding by suspension accounting for limited particle retention, controlled by the torque sum, allows for exact solution under the assumptions of constant filtration coefficient and porosity. The explicit formulae permit the calculation of the model parameters (maximum retention concentration, filtration and formation damage coefficients) from the history of the pressure drop across the core during suspension injection. The values for maximum retention concentration, as obtained from two coreflood tests, have been matched with those calculated by the torque balance on the micro scale.     

Controlled pendulum on a movable base

A plane motion of a multilink pendulum hinged to a movable base (a wheel or a carriage) is considered. The control torque applied between the base and the first link of the pendulum is independent of the base position and velocity and is bounded in absolute value. The coordinate determining the base position is cyclic. The mathematical model of the system permits one to single out the equations describing the pendulum motion alone, which differ from the well-known equations of motion of a pendulum with a fixed suspension point both in the structure and in the parameters occurring in these equations. The phase portrait of motions of a control-free one-link pendulum suspended on a wheel or a carriage is obtained. A feedback control ensuring global stabilization of the unstable upper equilibrium of the pendulum is constructed. Time-optimal control synthesis is outlined.     

Non-Newtonian end effects in falling ball viscometry of concentrated suspensions

In a Newtonian fluid contained in a cylinder, a small ball initially at rest released just below the surface would accelerate to achieve a steady-state velocity within one cylinder diameter. After traversing the center section of the cylinder, the ball would begin slowing down within one cylinder diameter of the bottom. This behavior is also observed in suspensions where the size of the suspended particles is small relative to the containing cylinder. However, in concentrated suspensions of larger suspended particles, balls released near the upper surface travel faster than the steady state velocity. In addition, the length of the upper surface end effect, where the falling ball decelerates to the steady state velocity, and the lower end effect zone, where the ball decelerates to rest at the bottom, is many times longer than in a Newtonian single-phase liquid. These non-Newtonian end effects are reduced if the suspended particles are polydisperse in their size distribution.     

The rheology of suspensions of porous zeolite particles in polymer solutions

We present data and predictive models for the shear rheology of suspended zeolite particles in polymer solutions. It was found experimentally that suspensions of zeolite particles in polymer solutions have relative viscosities that dramatically exceed the Krieger–Dougherty predictions for hard sphere suspensions. Our investigations show that the major origin of this discrepancy is due to the selective absorption of solvent molecules from the suspending polymer solution into zeolite pores. The effect raises both the polymer concentration in the suspending medium and the particle volume fraction in the suspension. Consequently, both the viscosity of the polymer solution and the particle contribution to the suspension viscosity are increased. We propose a predictive model for the viscosity of porous zeolite suspensions by incorporating a solvent absorption parameter, α, into the Krieger–Dougherty model. We experimentally determined the solvent absorption parameter by comparing viscosity data for suspensions of porous and nonporous MFI zeolite particles. Our results are in good agreement with the theoretical pore volume of MFI particles.     

Mathematical model of deep-bed filtration of a two-component suspension through a porous medium

A model of deep-bed filtration of a two-component suspension through a porous medium with formation of two types of the deposit which have different structures and properties is constructed. The influence of the parameters of fluid and particle flux densities which determine mass transfer between different components of the suspension and deposits on the filtration characteristics and properties of the resulting deposits is estimated on the basis of numerical experiments for the suspensions with contrast particle fractions.     

Kinetic theory of interacting particles in dilute suspensions of mineral waste

A kinetic theory of interacting spherical particles in dilute suspension is developed which results in a Boltzmann transport equation. This equation is solved in the relaxation time approximation to calculate the settling velocity of fine particulates in the steady state. The theory is applied to the suspended Jamaican bauxite waste and kaolinite particles. The experimental settling velocity compares well with the calculated values at low concentrations. This treatment can form the basis for a more rigorous theory applicable to denser systems and non-spherical particulates.     

Calculation of the motion of two-component media

The one-dimensional motion of a viscous incompressible liquid in which particles are suspended is described by the mathematical model used in [1], Two examples are discussed: the precipitation of particles from the suspension, and a boiling layer. The results are presented in the form of graphs.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 102–108, November–December, 1973.     

The sedimentation of polydisperse suspensions in vessels having inclined walls

A theory is presented for describing the sedimentation of polydisperse suspensions in two-dimensional channels having walls that are inclined to the vertical. The theory assumes that the flow is laminar and that the suspension consists of spherical beads having small particle Reynolds numbers. The suspension may consist of either $\text{N}$ distinct species of particles or of a continuum of particle sizes and densities. For the sake of simplicity, the analysis is mostly confined to the case in which the hindered settling velocity of each particle is given by its Stokes settling velocity multiplied by a function of the total local solids concentration. Under these conditions, results are developed that are useful for the design of either batch or continuous settling devices. Experimental observations were found to be in good agreement with the predictions of the present theory.     

A Novel Workflow to Model Permeability Impairment through Particle Movement and Deposition in Porous Media

This article presents a practical transfer function type solution to a complex problem in which variations in a number of parameters can be taken into account. A new mathematical model, which is based on mass balance transfer function of particles movement/retention in porous media, has been derived. It is used to predict permeability reduction as a function of time. The linear forms as well as the radial forms of the model are described. Although the differential equations derived are similar to the general form of diffusion–convection equations, the marked difference is the suitability of the model, for being applied for variation of parameters, such as particle concentration in the fluid, injection rate, density of solid particles, against the depth and time of invasion. This transfer function has been solved, and the results of the simulation run agree reasonably well with the experimental damage data obtained in laboratory. Owing to its simplicity, this model is more practical to describe permeability reduction for the flow of suspended particles in porous media.     

Quasistationary Sedimentation with Adsorption

Models for the sedimentation of particles suspended in solution are proposed that take into account mass transfer between the liquid fraction of the solution and the particles. The structure and velocity of the concentration wave describing the upward extension of the zone with a high concentration of the solid phase are studied using a kinematic model with the Froude number as a small parameter. It is established that the concentration-wave velocity becomes lower if the sorption parameter is set equal to zero, i. e., if the sorption properties of the suspended particles are ignored.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 66–77, July–August, 2005.